A lot of answers so far have been from other LDS scripture, but not much from the Book of Mormon itself. I wanted to provide further reference that even the people in the Book of Mormon asked this question:
Alma 39:17-19 (~74 B.C.)
17 And now I will ease your mind somewhat on this subject. Behold, you marvel why these things should be known so long beforehand. Behold, I say unto you, is not a soul at this time as precious unto God as a soul will be at the time of his coming?
18 Is it not as necessary that the plan of redemption should be made known unto this people as well as unto their children?
19 Is it not as easy at this time for the Lord to send his angel to declare these glad tidings unto us as unto our children, or as after the time of his coming?
The first three prophets in the Book of Mormon, Lehi, Nephi, and Jacob, were Jews that left Jerusalem to the Americas, and received revelation that helped them grow from their existing Judaic concept of a Messiah to more specific knowledge that the Messiah would be Christ and would eventually fulfil the law of Moses by His atonement.
They had visions of all aspects of Christ's life and mission, including His birth, baptism by John, His ministry, His suffering, death, and resurrection, and the names by which He would be known.
The revelation that the sinless Son of God would be baptized to keep the commandments of God cemented it as an essential practice (see 2 Nephi 31), but whether baptism was practiced by them prior to that is not mentioned in the Book of Mormon.
Lehi, Nephi, Jacob, Abinadi, and other Book of Mormon prophets expound extensively on the words of Old Testament prophets, especially Isaiah, to reinforce how their visions and revelations about Christ are consistent with what God had revealed in the past about the Messiah. They grow strongly in the belief that all the prophets have always testified of the Messiah Christ, and teach to believe in Him though He had not yet been born.
This is the same type of effort done by the early apostles in the New Testament to persuade the Jews that Christ was the promised Messiah, though they were basing it on their eyewitness accounts rather than vision. However we know that salvation doesn't come only from being an eyewitness, since "blessed are they that have not seen, and yet have believed. (John 20:29)"
The Book of Mormon teaches that ultimately a testimony of the redeeming power of Christ is confirmed by the power of the Holy Spirit, whether one lives before, during, or after the mortal life of Christ (Moroni 10:3-7).
Useful Book of Mormon Quotations on Revelations of Christ
Jacob 1: 5-6 (~544-421 B.C)
5 For because of faith and great anxiety, it truly had been made manifest unto us concerning our people, what things should happen unto them.
6 And we also had many revelations, and the spirit of much prophecy; wherefore, we knew of Christ and his kingdom, which should come.
2 Nephi 25:19 (~559-545 B.C.)
19 For according to the words of the prophets, the Messiah cometh in six hundred years from the time that my father left Jerusalem; and according to the words of the prophets, and also the word of the angel of God, his name shall be Jesus Christ, the Son of God.
2 Nephi 31 (~559-545 B.C.)
5 And now, if the Lamb of God, he being holy, should have need to be baptized by water, to fulfil all righteousness, O then, how much more need have we, being unholy, to be baptized, yea, even by water!
2 Nephi 25:26-27 (~559-545 B.C.)
26 And we talk of Christ, we rejoice in Christ, we preach of Christ, we prophesy of Christ, and we write according to our prophecies, that our children may know to what source they may look for a remission of their sins.
27 Wherefore, we speak concerning the law that our children may know the deadness of the law; and they, by knowing the deadness of the law, may look forward unto that life which is in Christ, and know for what end the law was given. And after the law is fulfilled in Christ, that they need not harden their hearts against him when the law ought to be done away.
Mosiah 3:13 (~124 B.C.)
13 And the Lord God hath sent his holy prophets among all the children of men, to declare these things to every kindred, nation, and tongue, that thereby whosoever should believe that Christ should come, the same might receive remission of their sins, and rejoice with exceedingly great joy, even as though he had already come among them.
Jarom 1:11 (~399-361 B.C.)
11 Wherefore, the prophets, and the priests, and the teachers, did labor diligently, exhorting with all long-suffering the people to diligence; teaching the law of Moses, and the intent for which it was given; persuading them to look forward unto the Messiah, and believe in him to come as though he already was. And after this manner did they teach them.
2 Nephi 33:10 (~559-545 B.C.)
10 And now, my beloved brethren, and also Jew, and all ye ends of the earth, hearken unto these words and believe in Christ; and if ye believe not in these words believe in Christ. And if ye shall believe in Christ ye will believe in these words, for they are the words of Christ, and he hath given them unto me; and they teach all men that they should do good.
If Stack Exchange markup allowed for collapsible quotations I'd include the actual verse text here, but as some of it is entire chapters I'll put links and summaries instead.
1 Nephi 1:5-6,19 (~600 B.C.)
Nephi begins the record of his people — Lehi sees in vision a pillar of fire and reads from a book of prophecy — He praises God, foretells the coming of the Messiah, and prophesies the destruction of Jerusalem — He is persecuted by the Jews.
1 Nephi 10:4-11 (~600-592 B.C.)
Lehi tells of the coming among the Jews of a Messiah, a Savior, a Redeemer — Lehi tells also of the coming of the one who should baptize the Lamb of God — Lehi tells of the death and resurrection of the Messiah.
1 Nephi 11 (~600-592 B.C.)
Nephi sees the Spirit of the Lord and is shown in vision the tree of life — He sees the mother of the Son of God and learns of the condescension of God — He sees the baptism, ministry, and crucifixion of the Lamb of God — He sees also the call and ministry of the Twelve Apostles of the Lamb.
1 Nephi 19:10,23 (588-570 B.C.)
The God of Israel will come six hundred years from the time Lehi left Jerusalem — Nephi tells of His sufferings and crucifixion
1 Nephi 20-22 are quotations from Isaiah 48-49 and how those writings apply with the revelations they have received.
2 Nephi 2:5-9 (~588-570 B.C.)
Redemption comes through the Holy Messiah
2 Nephi 6:8-9 (559-545 B.C.)
Jacob recounts Jewish history: The Babylonian captivity and return; the ministry and crucifixion of the Holy One of Israel
2 Nephi 9:4-6,21-24 (559-545 B.C)
The Atonement ransoms man from the Fall — The bodies of the dead will come forth from the grave, and their spirits from hell and from paradise — They will be judged — The Atonement saves from death, hell, the devil, and endless torment — The righteous are to be saved in the kingdom of God — Penalties for sins are set forth — The Holy One of Israel is the keeper of the gate.
2 Nephi 11:2-4,6 (~599-545 B.C.)
Jacob saw his Redeemer — The law of Moses typifies Christ and proves He will come.
2 Nephi 12-24 are quotations from Isaiah 2-14. The following chapters expound on how those writings apply with the revelations they have received.
2 Nephi 25:12-14,19-30 (~559-545 B.C.)
Nephi glories in plainness — Isaiah’s prophecies will be understood in the last days — The Jews will return from Babylon, crucify the Messiah, and be scattered and scourged — They will be restored when they believe in the Messiah — He will first come six hundred years after Lehi left Jerusalem — The Nephites keep the law of Moses and believe in Christ, who is the Holy One of Israel.
2 Nephi 31 (~559-545 B.C.)
Nephi tells why Christ was baptized — Men must follow Christ, be baptized, receive the Holy Ghost, and endure to the end to be saved — Repentance and baptism are the gate to the strait and narrow path — Eternal life comes to those who keep the commandments after baptism.
2 Nephi 32 (~559-545 B.C.)
Angels speak by the power of the Holy Ghost — Men must pray and gain knowledge for themselves from the Holy Ghost.
2 Nephi 33 (~559-545 B.C.)
Nephi’s words are true — They testify of Christ — Those who believe in Christ will believe Nephi’s words, which will stand as a witness before the judgment bar.
Jacob 1: 5-8 (~544-421 B.C)
Jacob and Joseph seek to persuade men to believe in Christ and keep His commandments
Jacob 4:4-7 (544-421 B.C.)
All the prophets worshiped the Father in the name of Christ — Abraham’s offering of Isaac was in similitude of God and His Only Begotten — Men should reconcile themselves to God through the Atonement
Jacob 7:10-12 (544-421 B.C.)
Sherem denies Christ, contends with Jacob, demands a sign, and is smitten of God — All of the prophets have spoken of Christ and His Atonement
Jarom 1:11 (~399–361 B.C.)
The Nephites keep the law of Moses, look forward to the coming of Christ, and prosper in the land — Many prophets labor to keep the people in the way of truth.
Mosiah 12 - 16 (~148 B.C.)
False priests quote the scriptures and pretend to keep the law of Moses — Abinadi begins to teach that salvation does not come by the law of Moses alone — God Himself will make an atonement and redeem His people.
Isaiah speaks messianically — The Messiah’s humiliation and sufferings are set forth — He makes His soul an offering for sin and makes intercession for transgressors — Compare Isaiah 53.
God redeems men from their lost and fallen state — Those who are carnal remain as though there were no redemption — Christ brings to pass a resurrection to endless life or to endless damnation.
Mosiah 3-6 (~124 B.C.)
King Benjamin continues his address — The Lord Omnipotent will minister among men in a tabernacle of clay — Blood will come from every pore as He atones for the sins of the world — His is the only name whereby salvation comes — Men can put off the natural man and become Saints through the Atonement — The torment of the wicked will be as a lake of fire and brimstone.
The Saints become the sons and daughters of Christ through faith — They are then called by the name of Christ.
... and the list goes on, but these are the earliest and most salient foundational scriptures in the Book of Mormon chronology that shows the people's growth from Messiah to Christ.